MembraneRestingPotentialandtheSodiumAnomaly
Belowisanimageofatypicalmembrane,wheretheintercellularregion(in)inside
thecellisfilledwithcytosol
(waterplusorganelles,ions
andproteins),andthe
extracellularregion(out)is
filledwithextracellularfluid
(mainlyionicsolutions).
RestingMembranePotential:
Theinterior(in)isusually
moreelectricallymorenegative
thantheexterior(out):
ΔV = Vin −Vout ≈ −60mV .
SodiumAnomaly:Inphysics,itiswellknownthatpositiveionsmovetomore
negativepotentialregion,andnegativeionsmovetomorepositivepotential.It
isthennotsurprisingthattherearemorepotassiumions,K+,outside
(extracellular)thaninside(intracellular),andtherearemorechlorideions,Cl− ,
insidethanoutside.Butitissurprisingthattherearemoresodiumions,Na+,
outside(positivepotential)thaninside(negativepotential).Thisiscalledthe
sodiumanomalyinPhillipNelson’stextbook.Databelowshowstheion
concentrationofatypicalliving(invivo)cell.
DataofmamaliancellMembraneRestPotential ΔV = −86mV ion z(valence) Intracellularci,in,mM Extracellularci,out,mM ViNersst,mV
K+ +1
140
+
Na +1
4
-
Cl
-1
4
NernstPotential:
4
140
120
?
?
?
InclassandinthetextbookwecalculatetheNernst’spotential Vi Nernst =
kBT ci,out
.
ln
ze
ci,in
= −90mV ;Na+, VNaNernst
= 89mV ;Cl-, VClNernst
= −86mV .Note
Forabovedata:K+, VKNernst
+
+
−
thattheyaredifferentthanthemembranepotential ΔV = −86mV .Onthenextpage,
Iwillillustratewhythisisasignificantpoint.
Nernst Relation
Consider the distribution of charge in the presence of a voltage (battery)
Considerthedistributionofchargeinthepresenceofavoltage(battery):
a
NB: q has sign, here
it is positive
The energy difference between charges on the top and bottom plate is:
So in the presence of an applied voltage charges will redistribute in this way
Thephysicalinterpretationoftheaboveisthatinthepresenceofanapplied
voltageΔV ,theequilibriumchargedistributionwillbeasthediagramabovewith
OR, if there is a difference in charge distribution, there will be a voltage
morepositiveionsonthenegativebottomplate.Anotherinterpretationisthatif
thereischargedistributionasintheabovefigure,thentherewillbeavoltage
C
kT
ΔV = − B ln top .Sincethetopismorepositivethannegativeweidentifythetop
q
C bottom
asextracellular(out)andthebottomasintracellular(in),soforcellswehave
kT C
ΔV = − B ln out = −Vi Nernst .BasicallythepotentialΔV ,depictedaboveisnegativeof
q
C in
theNernstpotentialequation Vi Nernst .
InterpretationofNernstPotential:FromOhm’slawweknowthatanapplied
voltagewillinduceacurrent,ΔV = IR .InthediagramaboveyheNernstpotential
V Nernst calculatedfortheabovedistributionistheappliedvoltageneededtoprevent
i
aflowofcharge–i.e.acurrent.
Whathappensinlivingcells?Intheabovedata(previouspage)weseethatin
< ΔV
generaltheNernstpotentialisdifferentfromthemembranepotential, VKNernst
+
Nernst
Nernst
and VNa+ > ΔV ,but VCl − = ΔV .Thismeansthattheremustbealeakcurrent,as
willbeillustratedinthenextexample.
Ionic current:
IMPORTANTPOINTS:
1.Intheabovedefinitiontheflowofionsispositiveifitisfromtheinside(in)to
outside(out),butinclassnotesweassumethatpositiveflowisfromoutside(out)
toinside(in),sotheequationshouldbeforourcourseandassignment:
I = Agi Vi Nernst − ΔV = Vi Nernst − ΔV / Ri ,whereAistheeffectivecrosssectionarea,
i
and Ri = Agi istheresistanceoftheioni(Na+,K+,Cl−).
2.Notethattheconductanceis G = 1 / R andtheconductanceperunitareais
g = G / A .Henceg = AG = A / R ,whichishowwederivedtheaboveequation.
3.FornegativeionssuchasCl−,theionflowdirectionisreversedasdefinedabove.
= ΔV isunusal.
4.Ingeneral, ΔV ≠ Vi Nernst .Thedataabove, VClNernst
−
(
) (
)